
with the direction of the dipole specifically chosen as \(\mathbf {e}=\mathbf {v}_0/|\mathbf {v}_0|\) , and the value of \(|\mathbf {v}_0|\)  is estimated as \(369.1 \pm 0.9~\mathrm {km}~\mathrm {s}^{-1}\)  in the CMB frame [1]}. Thus, a direct measurement of \(H(z)\)  can be given by the dipole anisotropy of luminosity distance. The error of \(H(z)\)  is related to the error of the luminosity distance, and can be estimated as [2]}, [3]}, [4]}
\(\frac{\Delta H(z)}{H(z)}=\frac{\Delta d_{L}^{(1)}(z)}{d_{L}^{(1)}(z)}=\sqrt{3}\left[\frac{d_{L}^{(1)}(z)}{d_{L}^{(0)}(z)}\right]^{-1}\left[\frac{\Delta d_{L}^{(0)}(z)}{d_{L}^{(0)}(z)}\right].\) 
